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Theorem uspgrloopedg 40734
 Description: The set of edges in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 40475) is a singleton of a singleton. (Contributed by AV, 17-Dec-2020.)
Hypothesis
Ref Expression
uspgrloopvtx.g 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
Assertion
Ref Expression
uspgrloopedg ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})

Proof of Theorem uspgrloopedg
StepHypRef Expression
1 uspgrloopvtx.g . . . 4 𝐺 = ⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩
21fveq2i 6106 . . 3 (Edg‘𝐺) = (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩)
3 snex 4835 . . . . 5 {⟨𝐴, {𝑁}⟩} ∈ V
43a1i 11 . . . 4 (𝐴𝑋 → {⟨𝐴, {𝑁}⟩} ∈ V)
5 edgaopval 25795 . . . 4 ((𝑉𝑊 ∧ {⟨𝐴, {𝑁}⟩} ∈ V) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
64, 5sylan2 490 . . 3 ((𝑉𝑊𝐴𝑋) → (Edg‘⟨𝑉, {⟨𝐴, {𝑁}⟩}⟩) = ran {⟨𝐴, {𝑁}⟩})
72, 6syl5eq 2656 . 2 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = ran {⟨𝐴, {𝑁}⟩})
8 rnsnopg 5532 . . 3 (𝐴𝑋 → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
98adantl 481 . 2 ((𝑉𝑊𝐴𝑋) → ran {⟨𝐴, {𝑁}⟩} = {{𝑁}})
107, 9eqtrd 2644 1 ((𝑉𝑊𝐴𝑋) → (Edg‘𝐺) = {{𝑁}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  ran crn 5039  ‘cfv 5804  Edgcedga 25792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-2nd 7060  df-iedg 25676  df-edga 25793 This theorem is referenced by: (None)
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